Snapping edges to surface not working in blender 2.8
Some variation on this?

Set  the origin of your object is at its base..
In Edit mode, assign the extruded vertices to a Vertex Group
In Object Mode, Switch on Snap, to Face, Align Rotation
Give the object a Shrinkwrap modifier, aimed at the Vertex Group, targeted on the surface.

Then, you can slide an object around on a surface, keeping contact.

In this case, the disc is up in Z, so I set the Shrinkwrap to negative Z projection. This may seem long-winded, but this can be a useful modelling routine.
You can take this further, making a flange, matching normals, etc., to make a Decal, but that's more than you want.